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At how many points is the function $f(x)=[x]$ discontinuous?
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The correct answer is:
Infinite
Given, $f(x)=[x]$ Let ' $c$ 'be any real number. $f$ is continuous at $x=c$ ifL.H.L. $=\mathrm{R}$.H.L. $=f(c)$
i.e., $\lim _{x \rightarrow c^{-}} f(x)=\lim _{x \rightarrow c^{+}} f(x)=f(c)$
L.H.L. $=\lim _{x \rightarrow c^{-}} f(x)=\lim _{x \rightarrow c^{-}}[x]=\mathrm{c}^{-1}$
R.H.L. $=\lim _{x \rightarrow c^{+}} f(x)=\lim _{x \rightarrow c^{+}}[x]=\mathrm{c}$
Since, L.H.L. $\neq$ R.H.L.
$f$ is discontinous for all $\mathrm{x} \in \mathrm{R}$.
So, $[x]$ is discontinuous at infinite points.
i.e., $\lim _{x \rightarrow c^{-}} f(x)=\lim _{x \rightarrow c^{+}} f(x)=f(c)$
L.H.L. $=\lim _{x \rightarrow c^{-}} f(x)=\lim _{x \rightarrow c^{-}}[x]=\mathrm{c}^{-1}$
R.H.L. $=\lim _{x \rightarrow c^{+}} f(x)=\lim _{x \rightarrow c^{+}}[x]=\mathrm{c}$
Since, L.H.L. $\neq$ R.H.L.
$f$ is discontinous for all $\mathrm{x} \in \mathrm{R}$.
So, $[x]$ is discontinuous at infinite points.
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